I have a tree in my back garden that I believe is a mulberry tree. It produces blackberry-like berries (which are delicious on my breakfast cereal!).

The problem is, it's growing so tall that some of it is drooping over the lawn and blocking all the sunlight. I'd like to prune it back.

Attached are some photos. You'll see how the trunk splits near the bottom - I'm wondering if I can just cut off the trunk on the right (that's the one that stretches over the lawn)? The trunk is about four inches across.

Will I kill the tree if I do this? Is there a right and wrong way to cut the trunk? Should I try pruning branches higher up instead?

You can remove the one on the right without killing the tree. You need to cut it above where you want the final trim to be .. I made a crude cutting sketch .. Cut a notch in it; and then above the notch make your falling cut. The notch will prevent the cut side from tearing away at the trunk as it falls . Then with the main part of that side gone , make the lower cut shown to remove the stub that was left in your removal cuts. Some recommend painting the cut wood with a roofing type tar to seal the wound . Others say not to. Search the pros and cons of it and decide whether to use it or not .. I have not used it when I pruned trees and I have not had any problems with the exposed wood healing properly .

With your final cut, make sure it is angled so that water runs off of the cut surface. Any horizontal surface cut will collect water and may make it more susceptible to decay. There is a correct way to prune ie. not leaving a stump or cutting too close to the main stem, etc. Research it before making that drastic step of pruning a large limb. Remember, where branches intersect and rub together, wounds can develop which leads to disease and rotting. I have read that pruning paint is unnecessary for healing the wound.